Spontaneous emergence of Josephson junctions in homogeneous rings of single-crystal Sr₂RuO₄

Sr₂RuO₄ is a prominent unconventional superconductor proposed to host a chiral order parameter with p-wave symmetry (k_x ± i k_y), supported by muon spin relaxation and polar Kerr measurements. Recent NMR work constrains the spin structure, suggesting that the d-vector is not parallel to the c-axis and allowing for possible chiral d-wave states. Chiral states are of interest due to potential Majorana bound states relevant for topological quantum computing. A key property of a chiral state is double degeneracy in the orbital degree of freedom, implying superconducting domains of opposite chirality and spontaneous edge currents. Despite indirect indications, such domains and edge currents have not been directly observed. Transport experiments using Ru inclusions suggest domain-related phenomena via proximity junctions, but the presence of Ru or strain can induce an extrinsic superconducting phase with enhanced T_c ≈ 3 K, believed to be non-chiral with a single-component order parameter. Here, the intrinsic multi-component phase has T_c ≈ 1.5 K, and the extrinsic single-component-like phase is characterized by T_c ≈ 3 K. Most prior studies used bulk crystals much larger than expected chiral-domain sizes (micron scale), and domain mobility has been inferred from switching noise. The energy cost of domain walls grows with area, making mesoscopic samples promising for stabilizing and probing chiral-domain walls (ChDWs). A ChDW can locally suppress the order parameter, acting as a Josephson junction. This work reports transport measurements on mesoscopic Sr₂RuO₄ rings fabricated from single crystals by focused ion beam (FIB). Homogeneous rings with sharp transitions around the intrinsic 1.5 K exhibit SQUID-like critical current oscillations with flux periodicity across the full temperature range below T_c, despite no artificial weak links. In contrast, rings in the extrinsic phase show only Little-Parks oscillations near the resistive transition and lack SQUID oscillations. Ginzburg–Landau simulations indicate that ChDWs naturally form and are pinned in the ring arms, providing intrinsic Josephson junctions.

The study builds on extensive literature regarding Sr₂RuO₄’s unconventional superconductivity and purported chiral order (e.g., muon spin relaxation and Kerr effect indicating time-reversal symmetry breaking). It acknowledges recent 17O NMR findings that challenge simple spin-triplet chiral p-wave interpretations, opening possibilities for chiral d-wave or helical states. Prior transport studies using Ru inclusions reported anomalous switching and domain dynamics, though Ru/strain can create an extrinsic 3-K phase with a likely single-component order parameter. Earlier mesoscopic experiments observed magnetoresistance oscillations and half-flux-quantum features, sometimes attributed to moving vortices or LP effects. Theoretical works on mesoscopic chiral p-wave superconductors predict multichiral ground states and topological transitions, suggesting that domain walls may form and act as weak links in confined geometries.

Materials and microfabrication: High-quality Sr₂RuO₄ single crystals were grown by the floating-zone method. Thin crystals (~0.7–1 µm thick) were obtained by crushing and immediately used to avoid air-induced degradation. Crystals were placed on SrTiO3 substrates and contacted either by high-temperature-cure silver paint (cured at 500 °C for 20 min) or by e-beam lithography and sputter-deposited gold (rings B and E). A 100-nm SiOx protective layer was deposited by e-beam evaporation before structuring. Microrings and four-wire contact geometries were sculpted using Ga⁺ FIB (30 kV, 20 pA). Representative geometries: Ring A rin ≈ 0.21 µm, rout = 0.55 µm, thickness ≈ 0.7 µm; Ring B rin = 0.30 µm, rout = 0.54 µm, thickness ≈ 0.7 µm. Sample quality was verified by R(T) before and after milling (no change in T_c), with high residual resistivity ratios (RRR) of 238 (A) and 177 (B). Transport measurements: DC transport was measured in a He refrigerator (down to 0.3 K). For resistance, current polarity was flipped to remove thermoelectric offsets; R = [V(I)+V(−I)]/(2I). Magnetotransport R(H) was measured with constant DC currents while sweeping axial magnetic field. Temperature stability was held to ~100 µK using an 80-Ω bypass resistor in parallel with the heater and tuned PID control, enabling resolution of LP-induced ΔT_c ~ 10 mK. Current–voltage characteristics V(I) and critical current I_c were measured at fixed T and H using triangular current sweeps at 2 mHz. Ic(H) was extracted over a range of temperatures below T_c. Magnetovoltage V(H) for fixed current directions was also recorded to analyze asymmetries and derive averaged R(H). Simulations: Time-dependent Ginzburg–Landau (TDGL) simulations for a chiral p-wave order parameter were performed on microring geometries with nanostructured transport leads (formalism per cited refs and Supplementary Information). Parameters were scaled by the temperature-dependent coherence length ξ(T) = ξ(0)/√(1−t) (with t = T/T_c). From critical field measurements, ξ(0) ≈ 66 nm (bulk-like). Simulations explored low-T (e.g., T ≲ 0.5 T_c) and near-T_c regimes for rings with dimensions matching A and B, mapping least-energy domain configurations as functions of ξ/r_in and ξ/r_out, and in axial magnetic field. A phase diagram of least-energy states (Meissner vs. configurations with 1, 2, 3 ChDWs or extended ChDW) was compiled. Comparative measurements for extrinsic phase: A separate ring (Ring C, 2-µm-thick crystal) exhibited a broadened transition starting at ~2.7 K after microstructuring, indicative of the extrinsic 3-K-like phase (RRR ≈ 129). Magnetoresistance near the transition was compared with simulated LP oscillations using measured ring dimensions and standard LP expressions for ΔT_c(H), assuming R(T) shifts horizontally by ΔT(H) without shape change. Field-dependent I_c(H) was also measured at lower T for A, B, and C.

- Observation of intrinsic SQUID-like behavior: Rings A and B (intrinsic phase, sharp transitions near ~1.5 K) exhibit pronounced critical current oscillations I_c(H) with period corresponding to one superconducting flux quantum through the ring area. The interference pattern shows cusp-shaped minima indicative of two symmetric parallel Josephson junctions. Oscillations are present from the onset of superconductivity down to the lowest temperatures measured and maintain their overall shape across temperature. - Temperature robustness: The amplitude of I_c(H) oscillations is essentially temperature independent (e.g., ΔI_c ≈ 12 µA at both 0.78 K and 1.50 K for Ring A), inconsistent with mechanisms based on circulating persistent currents or constriction weak links. - Geometric and material parameters: Ring A: r_in ≈ 0.21 µm, r_out = 0.55 µm, thickness ≈ 0.7 µm, RRR = 238; Ring B: r_in = 0.30 µm, r_out = 0.54 µm, thickness ≈ 0.7 µm, RRR = 177. Estimated ξ(0) ≈ 66 nm (bulk-like). For Ring B, the slightly larger inner radius yields a slightly shorter I_c(H) period, consistent with fluxoid quantization. - GL simulations: TDGL shows that in mesoscopic rings, chiral-domain walls (ChDWs) spontaneously form and are pinned in the arms, creating two parallel weak links where the order parameter is suppressed over a length ~ξ. At low T, two discrete ChDWs separate domains; near T_c, an extended coexistence region across the arms acts as weak links. Axial magnetic field displaces ChDWs but they remain in the arms due to confinement. A phase diagram versus ξ/r_in identifies regimes: Meissner (mono-domain) for wider arms and various multi-domain states for narrower arms, with the experimental rings expected to host ChDWs over most of the sub-T_c range. - Contrast with extrinsic phase and LP effect: In Ring C (extrinsic 3-K-like phase; onset ~2.7 K), magnetoresistance near the transition exhibits Little–Parks oscillations with period and amplitude matching simulations based on ring geometry and ΔT_c(H) expectations (ΔT_c ~ 10 mK). Crucially, no I_c(H) oscillations are observed at lower T in such extrinsic-phase rings; I_c(H) decays monotonically with field. - Exclusion of alternative explanations: The observed patterns cannot be explained by (i) modulation by circulating persistent currents (would yield sawtooth-like modulation with strong T dependence via L_K), (ii) current-driven moving vortices (large oscillations only near T_c), (iii) geometrical constriction junctions (would show strong T-dependent CPR and V–I hysteresis at low T), or (iv) accidental proximity/tunnel junctions (would induce extrinsic 3-K phase or contradict high metallicity and sharp single transitions).

The persistence of symmetric SQUID-like I_c(H) oscillations with fluxoid periodicity across the entire superconducting temperature range, together with negligible V–I hysteresis and temperature-invariant oscillation shape and amplitude, is consistent with two parallel Josephson weak links of intrinsic origin. TDGL simulations support a mechanism in which chiral-domain walls in a multicomponent order parameter are stabilized and pinned within the mesoscopic ring arms, locally suppressing the order parameter over ~ξ and forming effective junction barriers. The patterns’ symmetry implies approximately balanced junctions in opposite arms, consistent with domain-wall formation at equivalent positions. The confinement-induced pinning explains the insensitivity to cooling history (ZFC vs FC) and the robustness of oscillations. The absence of I_c(H) oscillations in rings dominated by the extrinsic 3-K-like phase, which instead show LP oscillations near T_c, underscores that the Josephson junctions are tied to the intrinsic multicomponent (1.5-K) phase. Alternative mechanisms (circulating currents, moving vortices, constrictions, accidental proximity/tunnel junctions) are inconsistent with the data regarding waveform, temperature dependence, and lack of hysteresis. While framed in terms of chiral p-wave, the GL formalism and domain-wall mechanism apply similarly to chiral d-wave; thus, the transport signatures indicate domain walls between degenerate chiral/helical states but do not uniquely identify the exact pairing symmetry.

Mesoscopic Sr₂RuO₄ rings in the intrinsic (∼1.5 K) phase exhibit spontaneous SQUID-like interference of the critical current with flux quantum periodicity across the full sub-T_c range, despite having no artificial weak links. TDGL simulations reveal that chiral-domain walls stabilize in the confined ring arms, acting as two parallel intrinsic Josephson junctions. Rings predominantly in the extrinsic (3-K-like) phase lack such junctions and display only Little–Parks oscillations near the transition, with no I_c(H) oscillations at lower temperatures. These results strongly support the existence of superconducting domain walls intrinsic to the multicomponent order parameter in Sr₂RuO₄ and demonstrate the power of combining mesoscopic structuring with order-parameter simulations to probe domain physics. Future work should aim to directly manipulate and image ChDWs, quantify their current–phase relations, and develop experiments that discriminate between candidate chiral p-wave, chiral d-wave, and helical states.

The transport measurements do not uniquely distinguish between chiral p-wave, chiral d-wave, or helical degenerate states; similar domain-wall phenomenology can arise in these cases. The evidence for domain walls is indirect (via transport and simulations) rather than direct imaging. Some author affiliation notes suggest possible sample-to-sample variation or fabrication-induced strain; while extrinsic-phase devices were identified and contrasted, residual strain effects in intrinsic-phase devices cannot be fully excluded. The precise microscopic structure and current–phase relation of the domain-wall junctions were not resolved and remain for future study.